Arithmetic and Dynamical Degrees on Abelian Varieties
Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 1, pp. 151-167.

Let φ:XX be a dominant rational map of a smooth variety and let xX, all defined over ¯. The dynamical degree δ(φ) measures the geometric complexity of the iterates of φ, and the arithmetic degree α(φ,x) measures the arithmetic complexity of the forward φ-orbit of x. It is known that α(φ,x)δ(φ), and it is conjectured that if the φ-orbit of x is Zariski dense in X, then α(φ,x)=δ(φ), i.e. arithmetic complexity equals geometric complexity. In this note we prove this conjecture in the case that X is an abelian variety, extending earlier work in which the conjecture was proven for isogenies.

Soit φ:XX une application rationnelle dominante d’une variété lisse et soit xX, tous deux définis sur ¯. Le degré dynamique δ(φ) mesure la complexité géométrique des itérations de φ, tandis que le degré arithmétique α(φ,x) mesure la complexité arithmétique de la φ-orbite de x. Il est connu que α(φ,x)δ(φ), et il est conjecturé que si la φ-orbite de x est Zariski dense dans X, alors α(φ,x)=δ(φ). Dans cette note, nous prouvons cette conjecture dans le cas où X est une variété abélienne, étendant des travaux antérieurs où la conjecture a été prouvée pour les isogénies.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.973
Classification: 37P30,  11G10,  11G50,  37P15
Keywords: dynamical degree, arithmetic degree, abelian variety
Joseph H. Silverman 1

1 Mathematics Department, Box 1917 Brown University, Providence, RI 02912, USA
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Joseph H. Silverman. Arithmetic and Dynamical Degrees on Abelian Varieties. Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 1, pp. 151-167. doi : 10.5802/jtnb.973. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.973/

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